Section: New Results

Robust state estimation (Sensor fusion)

This research is the follow up of Agostino Martinelli's investigations carried out during the last five years, which are in the framework of the visual and inertial sensor fusion problem and the unknown input observability problem.

Visual-inertial structure from motion

Participants : Agostino Martinelli, Alexander Oliva, Alessandro Renzaglia.

During this year, we have obtained the full analytic solution of the cooperative visual inertial sensor fusion problem in the case of two agents, starting from the closed-form solution obtained in the last years (this latter solution will be published on the journal of Autonomous Robots [76]). Additionally, we also validated this solution with real experiments and in particular we showed that the analytic solution significantly outperforms our previous closed-form solution in [76]. The new analytic solution has been accepted for publication by the IEEE Robotics and Automation Letters [13].

Specifically, we obtained the analytic solution of the problem by first proving that, this sensor fusion problem, is equivalent to a simple polynomial equations system that consists of several linear equations and three polynomial equations of second degree. The analytic solution of this polynomial equations system was easily obtained by using an algebraic method (developed by Bernard Mourrain, the leader of AROMATH at Inria Sophia Antipolis). The power of the analytic solution is twofold. From one side, it allows us to determine the relative state between the agents (i.e., relative position, speed and orientation) without the need of an initialization. From another side, it provides fundamental insights into all the theoretical aspects of the problem. During this year, we focused on the first issue. Our next objective is to exploit the analytic solution to obtain basic structural properties of the problem.

Unknown Input Observability

Participant : Agostino Martinelli.

The Unknown Input Observability problem (UIO) in the nonlinear case was an open problem since the sixties years, when it was solved only in the linear case. In the last five years, I have obtained its general analytic solution. The mathematics apparatus necessary to obtain this solution is very sophisticated and is based on Ricci calculus, borrowed from theoretical physics. On the other hand, this mathematics can be avoided in the case of driftless systems and characterized by a single unknown input.

All the results (i.e., in the general case that also accounts for a drift and more than one unknown input) are fully described in a book available on ArXiv (arXiv:1704.03252).

During this year, my effort was devoted to make the analytic derivation of the solution palatable for a large audience (in particular, without knowledge of Ricci calculus). Hence, I focused on the simple case of a single unknown input and without drift. This solution has been published on a full paper on the IEEE Transaction on Automatic Control [75].

Regarding the general case available on ArXiv (arXiv:1704.03252), I was invited by the SIAM to write a book, palatable for a large audience. The scope of writing this book, is to present to the control theory and information theory communities a very powerful mathematics framework borrowed from theoretical physics. This could provide the possibility of revisiting many aspects of the control and information theory and bring new fundamental results, open new research domains etc. In this sense the book could be the kick-off of a new season of research in control and information theory. This will be the objective of the next years.